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COMMENTARY FOR THE AAT MATHEMATICS GROUP REPORT, February 2004:
1. There is agreement by the AAT Mathematics Group that the AAT mathematics
document include the following:
1. 18 Outcomes, which are directly related to the basic topics
comprising the 3-course sequence Calculus I, II, and III, as well as lower-division Linear
Algebra.
2. Either of the following two tracks for the science component:
A. Track 1: 2-course Calculus-based Physics I-II
B. Track 2: 2-course Algebra-based Physics I-II or 2-course Chemistry I-II.
Among the 4-year institutions the assignment of tracks are as follows:
Bowie State University: Track 1
Coppin State College: Track 1
Frostburg State University: Track 1 or Track 2
Hood College: Track 1 or Track 2
Loyola College: Track 1
Morgan State University: Track 1
College of Notre Dame: Track 1 or Track 2
Mt. St. Mary's College: Track 1 or Track 2
Salisbury University: Track 1 or Track 2
St. Mary's College: Track 1 or Track 2
Towson University: Track 1
UMBC: Track 1
UMCP: Track 1 or Track 2
UMES: Track 1
Washington College: Track 1 or Track 2
Note: As for Linear Algebra, this lower-division course should transfer as such
to any public 4-year institution statewide, irrespective of the numbering at
the 4-year institution.
Respectfully submitted,
Dan Symancyk and Denny Gulick, co-chair
1
Secondary Associates of Arts in Teaching
Content Area: Mathematics
Standard: Linear Algebra, Calculus
Teacher candidates will be able to:
Outcomes
1. Understand matrices
and their applications
Indicators
a. Perform matrix operations (addition,
subtraction, multiplication, scalar
multiplication, and transposition);
demonstrate knowledge of properties
of the above operations.
b. Solve linear systems using elementary
row operations.
c. Have a working knowledge of inverse
matrices (existence of, calculation of,
and use in solving systems of
equations)
d. Have a working knowledge of
determinants (existence of, operations
on, calculation of, and properties of)
and relationship to geometry.
Assessment Type
Sample Assessment Tasks


Extended response
Given a square matrix, use
elementary row operations to
determine whether the inverse
exists, and if so, find the inverse.
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2. Understand vectors and
vector spaces and their
applications.
a.
b.
c.
d.
e.
f.
3. Understand linear
transformations and their
applications
a.
b.
c.
d.
e.
Recognize vector spaces and
subspaces of vector spaces.
Operate on vectors in Rn and describe
the geometrical interpretations of
those operations.
Identify when two or more vectors are
linearly independent and/or span.
Recognize a basis for a vector space,
and determine the dimension of a
vector space.
Identify orthogonal vectors.
Represent vectors with respect to
different bases.
Recognize linear transformations and
their connection to matrices.
Determine null space, image, and
rank of a linear transformation.
Determine whether two vector spaces
are isomorphic.
Find and analyze eigenvalues,
eigenvectors, characteristic
polynomials, and eigenspaces.
Perform the diagonalization process
on a square matrix.

Extended response

Given a subset of a vector space,
determine whether the subset is a
subspace.

Extended response

Given the vector space of
polynomials of degree less than
or equal to four, determine the
matrix of the linear
transformation given by
differentiation with respect to the
standard basis.
3
4. Use technology to
assist with calculations
and explorations
a.
b.
c.
Having demonstrated manual
computational skills, use technology
to assist with those computations
where appropriate (e.g., operations on
matrices, calculating determinants).
Explore use of technology in complex
“real world” computations (e.g.,
solving linear systems of equations,
finding L-U factorization of matrices,
and finding eigenvalues of matrices).
Use technology to expand
explorations of linear algebra.


Projects
Extended response

Use technology to explore the
images of the unit square in the
first quadrant under several
linear transformations.
5. Identify the properties
of basic classes of
functions. (Here
"functions" are algebraic,
inverse, exponential
[including hyperbolic],
logarithmic,
trigonometric.)
6. Calculate the limits of
functions.
a.
Verify properties of symmetric,
inverse, and composite functions for a
collection of algebraic, exponential,
logarithmic, and trigonometric
functions.

Short answers, extended
responses, graphs,
quizzes, tests

Given the graph of a function,
tell whether it is of exponential,
logarithmic, polynomial, or
trigonometric type.
a.

Evaluate a limit using l’Hôpital’s
Rule
a.
Short answers, extended
responses, graphs,
quizzes and tests.
Short answers, extended
responses, graphs,
quizzes and tests

7. Analyze continuity of a
function
Analyze problems using the
Squeezing Theorem, one-sided limits,
infinite limits, l’Hôpital’s Rule.
Identify continuity and piecewise
continuity of functions and analyze
properties of continuity through the
Intermediate Value Theorem.

Use the Intermediate Value
Theorem to show that the range
of the sine function contains all
numbers in the interval [-1, 1].

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8. Find the derivatives of
functions numerically,
algebraically, and
graphically.
a.
b.
9. Apply the derivative to
diverse situations.
a.
b.
10. Calculate definite
integrals and find
indefinite integrals.
a.
b.
Calculate the derivative of a function
(using basic rules of differentiation,
including the chain rule and implicit
differentiation) and use it to find the
slope, tangent, higher derivatives.
Estimate approximate values of
functions (with technology), and find
the relation between the derivative of
a function and its inverse.
Apply the derivative to find related
rates, velocity and acceleration from
position, properties of graphs of
functions (including relative extrema,
asymptotes, concavity), solutions of
maximum and minimum problems,
and exponential growth and decay.
Explain the uses of Rolle’s Theorem
and the Mean Value Theorem.
Apply Riemann Sums, the
Fundamental Theorem of Calculus,
algebraic and trigonometric
substitutions, integration by parts, and
partial fractions to find integrals.
Estimate values of integrals by means
of Simpson’s Rule (with technology).
Explain why differentiation and
integration are inverse processes, and
indicate the historical roles of Newton
and Leibniz for the calculus.

Short answers, extended
responses, graphs,
quizzes and tests

Determine the values (if any)
where the line tangent to a given
third degree polynomial is
horizontal.

Short answers, extended
responses, graphs,
quizzes and tests

Find the maximum volume of a
right circular cylinder that is
inscribed in a given sphere.

Short answers, extended

responses, graphs
quizzes, tests short essays
Calculate the area of the region
bounded above by the sine
function, below by the x-axis,
between x = 0 and x = .
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11. Solve applied
problems related to
integration.
a.
12. Analyze the
convergence or
divergence of sequences
and series.
a.
b.
c.
d.
13. Graph and analyze
polar equations,
parametric equations, and
conic sections.
a.
b.
c.
Using integration, find solutions to
problems involving area, volume,
surface area, work, moments, and
length of a curve, as well as position
and velocity from known
acceleration.
Use convergence properties of
sequences to determine the
convergence or divergence of a given
sequence.
Use the convergence tests (nth term
test, integral test, ratio test, alternate
series test) to determine the
convergence or divergence of given
series.
Find the power series and Taylor
series for given functions with the
Lagrange Remainder Formula.
Apply Taylor’s Theorem, absolute
convergence to power series, and find
the radius of convergence of a power
series.
Analyze functions given in polar form
or in parametric form.
Analyze rectangular forms of conic
sections.
Calculate lengths and areas related to
polar and parametric functions.

Short answers, extended
responses, graphs,
quizzes and tests

Find the area of the region
formed by a given ellipse.

Short answers, extended
responses, graphs,
quizzes and tests

Find the Taylor series for the
sine function, and determine the
radius of convergence of the
Taylor series.

Short answers, extended
responses, graphs,
quizzes and tests

Discuss the properties of the
cycloid.
6
14. Solve elementary
differential equations.
a.
b.
15. Explain properties of
vectors and vector-valued
functions.
c.
a.
b.
16. Apply differentiation
rules to various
multivariable functions.
Identify these properties
of quadric surfaces.
a.
b.
c.
17. Evaluate multiple
integrals.
a.
b.
Explain basic definitions relative to
differential equations and solve
separable differential equations.
Find approximate solutions, for
example, using Euler’s method.
Sketch a solution given a slope-field.
Calculate dot products, cross
products, distances between points
and lines in space.
Find derivatives, tangents, normals,
curvature for parameterized curves
(using technology).
Find directional derivatives,
gradients, tangent planes, and
approximations (using technology) by
means of partial derivatives.
Find extreme values of multivariable
functions, including the use of
Lagrange multipliers.
Describe geometric properties of
multivariable functions, including
level curves and quadric surfaces.
Evaluate double and triple integrals
using rectangular, cylindrical and
spherical coordinates, as well as
change of variables.
Find volumes, mass and moments of
objects in space.

Short answers, extended
responses, graphs,
quizzes and tests

Solve the differential equations
for the exponential growth and
decay.

Short answers, extended
responses, graphs,
quizzes and tests

Find the distance between a
given point and a given line in
space.

Short answers, extended
responses, graphs,
quizzes and tests

Find the plane tangent to the
graph of a given paraboloid.

Short answers, extended
responses, graphs,
quizzes and tests

Find the volume of the solid
region that lies inside a given
cone and given sphere
7
18. Explain properties of
vector fields and evaluate
various vector field
derivatives and integrals.
a.
b.
Explain and calculate the divergence,
gradient and curl of a given function.
Evaluate line integrals, and surface
integrals by means of the
Fundamental Theorem of Line
Integrals, Green’s Theorem, Stokes’s
Theorem and the Divergence
Theorem.

Short answers, extended
responses, graphs,
quizzes and tests

Show that a given line integral is
independent of path.
The above outcomes are included in the following courses:
Calculus I , II, and III (for science, engineering, and mathematics majors)
Linear Algebra (sophomore level)
To meet their general education science requirements, mathematics education majors must complete Calculus-based Physics I and II for Track I, or
either Algebra-based Physics I and II or Chemistry I and II for Track II.
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